2D TRANSFORMATIONS

Introduction

Supposing we taking a picture from a camera, when the picture is loaded in the computer screen, we feel that what if the picture was taken from a different angle, say a bit closer to the object, or from some height or even from some different direction. But most of the times it’s not possible to have the same sight in front of us again. Hence the program called Geometric Transformation was introduced where the image can be altered as per our wish without altering the originality of the image or the object.

2D Geometric Transformationof images includes changing the geometrical dimensions of theimageand thus improving the output image. It is being widely used in computer graphics these days. It actually forms a mapping function between the input image and the resultant image and makes a geometric relation between the distorted point of the input image and the same corrected point of the output image. The pixels’ coordinates has to be precisely known to the designer in order to apply the Geometric Transformation to the image and correct the distortions in it.

2D Geometric Transformation Techniques

Geometric Transformationinvolve many methods to improve the image, some of them are being discussed here:

· Translation-Translationmeans moving the coordinates of the image to a new position by adding translation may result in a better form of image and also change the texture. But one thing should be kept in mind that every point of the image should be increased or decreased by same amount of points.Transformation can be represented in the form of formulae.L’=L+T;Where L’ is the newly formed line or image, L denotes the original image and T is the Translation points that are added to the coordinates of the points of the image.

· Scaling-Scalingmeans changing the overall size of the image by moving the point scale up or down and getting the desired size. Scaling is of two types,Uniform Scaling-In Uniform Scaling the same amount of points are added or deducted from both x and y points or axes. HereSx= SyWhere S denotes the change in the respective axis.Non-Uniform-In Non Uniform Scaling different amount of points is added or deducted from the x and y points or axes.Sx≠ Sy.Where S denotes the change in the respective axis.

-Effects Of Different Types Of Scaling-1.Sx, Sy< 1,The size of the image is reduced and it moves closer to the origin.2.Sx, Sy> 1,The size of the image is increased and it moves away from the origin.3.Sx= Sy,The size of the image does not change.

· Rotation-Rotatingthe image or the object on its axis by some degree to get a different view side of it is known as Rotation. That is adding some degree of the image. This degree is denoted by ‘Ѳ‘ . The equation thus obtained after altering the direction of image is:x’ = x cos (Ѳ) – y sin (Ѳ)y’ = x sin (Ѳ) + y sin (Ѳ)where x’ and y’ is the new coordinates of the altered image.

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